Two-dimensional Case Research Articles

Stability and convergence analysis of high-order numerical schemes with DtN-type absorbing boundary conditions for nonlocal wave equations

Abstract The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes to discretize the spatially nonlocal operator, and apply the explicit difference scheme to approximate the temporal derivative to achieve a fully discrete infinity system. After that, we construct the Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs), to reduce the infinite discrete system into a finite discrete system. To do so, we first adopt the idea in Du et al. (2018, Commun. Comput. Phys., 24, 1049–1072) and Du et al. (2018, SIAM J. Sci. Comp., 40, A1430–A1445) to derive the Dirichlet-to-Dirichlet (DtD)-type mappings for one- and two-dimensional cases, respectively. We then use the discrete nonlocal Green’s first identity to achieve the discrete DtN-type mappings from the DtD-type mappings. The resulting DtN-type mappings make it possible to perform the stability and convergence analysis of the reduced problem. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach.

On the duality of unsteady stretching surface flow: the sheet and disk cases

This communication is a continuation of our previously published works on the duality of solution of stretching/shrinking surface flows. In this regard, a huge volume of literature is already available advocating that the duality of the solution is a salient feature of the shrinking surface flows, only. It is believed that for a shrinking surface flow the solution does not exist; requires sufficient wall suction for its existence; and is non-unique whenever it exists. On this basis, the shrinking surface flow is believed to be richer in physics by exhibiting more non-linear phenomena. Because of such facts, the available literature has developed a significant miSunderstanding about the shrinking surface flows. It has been established that the shrinking surface flows constitute a superior class of viscous flows instigated by the motion of a continuous surface. However, we think that there is nothing special with the shrinking surface flows, at all. All the fascinations attributed to the shrinking surface flows can be seen for the stretching surface flows, too. The reason behind the appearance of such fascinating features is the retarded nature of the ongoing boundary-layer flow. A retarded boundary-layer can be assisted in a variety of ways; and an assisted, retarded, boundary-layer exhibits a non-unique behavior. In this study, the duality of the solution has been discussed for unsteady stretching surface flow by considering two-dimensional planner and axisymmetric cases corresponding to flat sheet and disk geometries. It is shown that in a stretching surface flow, it is possible to observe the duality of the solution not only in the presence of wall suction but also in the presence of wall injection velocity.

Improved hybrid Allen-Cahn phase-field-based lattice Boltzmann method for incompressible two-phase flows.

In this work we develop an improved phase-field based lattice Boltzmann (LB) method where a hybrid Allen-Cahn equation(ACE) with a flexible weight instead of a global weight is used to suppress the numerical dispersion and eliminate the coarsening phenomenon. Then two LB models are adopted to solve the hybrid ACE and the Navier-Stokes equations, respectively. Through the Chapman-Enskog analysis, the present LB model can correctly recover the hybrid ACE, and the macroscopic order parameter used to label different phases can be calculated explicitly. Finally, the present LB method is validated by five tests, including the diagonal translation of a circular interface, two stationary bubbles with different radii, a bubble rising under the gravity, the Rayleigh-Taylor instability in two-dimensional and three-dimensional cases, and the three-dimensional Plateau-Rayleigh instability. The numerical results show that the present LB method has a superior performance in reducing the numerical dispersion and the coarsening phenomenon.

Global well-posedness of a new class of initial-boundary value problem on incompressible MHD/MHD-Boussinesq equations

The global well-posedness of a new class of initial-boundary value problem on incompressible MHD equations in the bounded domain with the smooth boundary is studied. The existence of a class of global weak solution to the initial boundary value problem for two/three-dimensional incompressible MHD equation with the given pressure-velocity's relation boundary condition for the fluid field at the boundary and with one perfectly insulating boundary condition for the magnetic field at the boundary is obtained, and the global existence and uniqueness of the smooth solution to the corresponding problem in two-dimensional case for the smooth initial data is also established. The corresponding results are also extended to the two/three-dimensional incompressible MHD-Boussinesq equations with the density-velocity's relation boundary condition for the density.

Towards a novel EMHD dissipative stagnation point flow model for radiating copper-based ethylene glycol nanofluids: An unsteady two-dimensional homogeneous second-grade flow case study

A novel EMHD dissipative second-grade nanofluid flow model is proposed exclusively in this numerical inspection for radiating copper-based ethylene glycol nanofluids to reveal the dynamical and thermal aspects of the studied homogeneous mixture during its unsteady two-dimensional stagnation point flow towards a horizontal electromagnetic actuator. Based on admissible physical assumptions and authenticated experimental correlations, the governing PDEs and BCs are derived appropriately for the nanofluid flow problem under consideration. After numerous rearrangements and non-dimensionalization treatments, the resulting ODEs and BCs are handled computationally with the help of a robust GDQ algorithm under the parametric control of several influencing factors, whose strengthening magnitudes affect probably the flow control process and heat transport mechanism. In this context, it proved graphically that the nanoparticles’ loading process exhibits dissimilar dynamical and thermal impacts as compared with the influences of the nanoparticles’ diameter size. Besides, the resistive dynamical effect of the utilized electromagnetic actuator reinforces thermally the enhancing role of the thermal radiative heat flux and Joule’s heating process within the nanofluidic medium.

Graph neural networks for the prediction of aircraft surface pressure distributions

Aircraft design requires a multitude of aerodynamic data and providing this solely based on high-quality methods such as computational fluid dynamics is prohibitive from a cost and time point of view. Deep learning methods have been proposed as surrogate models to predict aerodynamic quantities, showing great potential at significantly reduced cost. However, most approaches rely on a structured grid or are tested only for two-dimensional airfoil cases with a few thousand nodes. During aircraft programs, unstructured grids with millions of nodes are routinely used to model industrial-relevant complex physical systems. Hence, further investigation is required to study the applicability and extension of deep learning methods to industrial cases. In this paper, we use a graph neural network approach applicable to unstructured grids and extend it for the task of predicting surface pressure distributions for complex cases involving several hundreds of thousand of nodes. We compare this approach with proper orthogonal decomposition combined with an interpolation technique and with two other deep learning approaches, namely, a coordinate-based multilayer perceptron for pointwise predictions and its extension using surface normals as additional inputs. Results are first presented for a two-dimensional airfoil case and then for the NASA Common Research Model transport aircraft with an underlying mesh consisting of around 500,000 surface points. The deep learning methods demonstrate in transonic flows the ability to capture shock location and strength more accurately. Furthermore, the proposed graph-based approach with the addition of more geometric information such as connectivity and surface normals seems to provide an additional boost in performance over the coordinate-based multilayer perceptron yielding more realistic pressure distributions.

Non-monotonicity of the first eigenvalue for the 3D magnetic Robin Laplacian

Previous works provided several counterexamples to monotonicity of the lowest eigenvalue for the magnetic Laplacian in the two-dimensional case. However, the three-dimensional case is less studied. We use the results obtained by Helffer, Kachmar, and Raymond to provide one of the first counterexamples in 3D. Considering the magnetic Robin Laplacian on the unit ball with a constant magnetic field, we show the non-monotonicity of the lowest eigenvalue asymptotics when the Robin parameter tends to +infty .

Retraction Note: Proof of the Jacobian Conjecture in the Two-Dimensional Case and Global Isochronous Centers of Polynomial Hamiltonian Differential Systems

The author has retracted this article. The main results of the work, namely Theorem 1 and the corollaries arising from it, have not been strictly proven: formulas (8) are valid only under the condition. Yet, even when the indicated condition is met, formulas (9) need to be proven. Because of this, this work in its current form should be recognized as incorrect. The author is grateful to Professor L.G. Makar-Limanov, who drew attention to the lack of correct proof of formula (8) and Professor V.V. Bakhtin, who pointed to other inaccurate statements in this article.

Well-posedness of a Navier–Stokes–Cahn–Hilliard system for incompressible two-phase flows with surfactant

We investigate a diffuse-interface model that describes the dynamics of viscous incompressible two-phase flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn–Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier–Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (respectively, local) strong solution in two (respectively, three) dimensions. In the two-dimensional case, we derive a continuous dependence estimate with respect to the norms controlled by the total energy. Moreover, we establish instantaneous regularization properties of global weak solutions for [Formula: see text]. In particular, we show that the surfactant concentration stays uniformly away from the pure states 0 and 1 after some positive time.

An iterative method based on Nesterov acceleration for identifying space-dependent source term in a time-fractional diffusion-wave equation

In this paper, we investigate an inverse problem of identifying a space-dependent source in a time-fractional diffusion-wave equation by using the final time data. Since the problem is ill-posed, we propose an iterative method based on the Nesterov acceleration strategy to deal with it. Two kinds of convergence rates for the regularized solution are given under both the a priori and a posteriori regularization parameter choice rules. Compared with the classical regularization methods, our method is easy to implement with a fewer number of iterations and can always yield the order optimal error estimates provided the iteration parameter is chosen large enough. Some numerical examples including one-dimensional and two-dimensional cases are presented to illustrate the validity and effectiveness of the proposed method.

Topology optimization for additive manufacturing with strength constraints considering anisotropy

Abstract Combining topology optimization and additive manufacturing (AM) is a promising approach to breaking through the limitations of conventional design and developing innovative structures with high performance. However, the unique manufacturing constraints in AM should be considered when developing the topology optimization algorithms for AM. Material anisotropy is one of the common characteristics of AM materials due to the layer-by-layer manufacturing techniques. This work proposes a topology optimization approach for AM with strength constraints considering anisotropy. The Hoffman failure criterion is adopted to represent the anisotropic strength behaviors of AM materials. Based on the Hoffman failure criteria and the p-norm measure aggregation function, a global strength constraint formulation is established. Under the framework of solid isotropic material with penalization, we develop a topology optimization methodology to minimize the structural weight or volume fraction subject to structural stiffness and strength constraints. Several two-dimensional (2D) or 3D numerical test cases are performed to validate the effectiveness and performance of the developed method. The results indicated that the proposed method could make full use of material properties by considering anisotropic strength. Besides, the topological optimization considering strength anisotropy could be combined with build direction optimization to further reduce the structural weight.

Deformed random walk: Suppression of randomness and inhomogeneous diffusion.

We study a generalization of the random walk (RW) based on a deformed translation of the unitary step, inherited by the q algebra, a mathematical structure underlying nonextensive statistics. The RW with deformed step implies an associated deformed random walk (DRW) provided with a deformed Pascal triangle along with an inhomogeneous diffusion. The paths of the RW in deformed space are divergent, while those corresponding to the DRW converge to a fixed point. Standard random walk is recovered for q→1 and a suppression of randomness is manifested for the DRW with -1<γ_{q}<1 and γ_{q}=1-q. The passage to the continuum of the master equation associated to the DRW led to a van Kampen inhomogeneous diffusion equation when the mobility and the temperature are proportional to 1+γ_{q}x, and provided with an exponential hyperdiffusion that exhibits a localization of the particle at x=-1/γ_{q} consistent with the fixed point of the DRW. Complementarily, a comparison with the Plastino-Plastino Fokker-Planck equationis discussed. The two-dimensional case is also studied, by obtaining a 2D deformed random walk and its associated deformed 2D Fokker-Planck equation, which give place to a convergence of the 2D paths for -1<γ_{q_{1}},γ_{q_{2}}<1 and a diffusion with inhomogeneities controlled by two deformation parameters γ_{q_{1}},γ_{q_{2}} in the directions x and y. In both the one-dimensional and the two-dimensional cases, the transformation γ_{q}→-γ_{q} implies a change of sign of the corresponding limits of the random walk paths, as a property of the deformation employed.

Symmetrized fractional total variation for signal and image analysis

We introduce and study a variational model for signal and image analysis based on Riemann–Liouville fractional derivatives. Both the one-dimensional and two-dimensional cases are studied. The model exploits a quadratic fitting data term together with both right and left Riemann–Liouville fractional derivatives as regularizing terms, with the aim of achieving an orientation-independent analysis.

Stability estimate for scalar image velocimetry

Abstract In this paper, we analyze the stability of the system of partial differential equations modelling scalar image velocimetry. We first revisit a successful numerical technique to reconstruct velocity vectors u {{u}} from images of a passive scalar field ψ by minimizing a cost functional that penalizes the difference between the reconstructed scalar field ϕ and the measured scalar field ψ, under the constraint that ϕ is advected by the reconstructed velocity field u {{u}} , which again is governed by the Navier–Stokes equations. We investigate the stability of the reconstruction by applying this method to synthetic scalar fields in two-dimensional turbulence that are generated by numerical simulation. Then we present a mathematical analysis of the nonlinear coupled problem and prove that, in the two-dimensional case, smooth solutions of the Navier–Stokes equations are uniquely determined by the measured scalar field. We also prove a conditional stability estimate showing that the map from the measured scalar field ψ to the reconstructed velocity field u, on any interior subset, is Hölder continuous.

A joint absorbing boundary for the multiple-relaxation-time lattice Boltzmann method in seismic acoustic wavefield modeling

Conventional seismic wave forward simulation generally uses mathematical means to solve the macroscopic wave equation, and then obtains the corresponding seismic wavefield. Usually, when the subsurface structure is finely constructed and the continuity of media is poor, this strategy is difficult to meet the requirements of accurate wavefield calculation. This paper uses the multiple-relaxation-time lattice Boltzmann method (MRT-LBM) to conduct the seismic acoustic wavefield simulation and verify its computational accuracy. To cope with the problem of severe reflections at the truncated boundaries, we analogize the viscous absorbing boundary and perfectly matched layer (PML) absorbing boundary based on the single-relaxation-time lattice Boltzmann (SRT-LB) equation to the MRT-LB equation, and further, propose a joint absorbing boundary through comparative analysis. We give the specific forms of the modified MRT-LB equation loaded with the joint absorbing boundary in the two-dimensional (2D) and three-dimensional (3D) cases, respectively. Then, we verify the effects of this absorbing boundary scheme on a 2D homogeneous model, 2D modified British Petroleum (BP) gas-cloud model, and 3D homogeneous model, respectively. The results reveal that by comparing with the viscous absorbing boundary and PML absorbing boundary, the joint absorbing boundary has the best absorption performance, although it is a little bit complicated. Therefore, this joint absorbing boundary better solves the problem of truncated boundary reflections of MRT-LBM in simulating seismic acoustic wavefields, which is pivotal to its wide application in the field of exploration seismology.

Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise

In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results.

Three-dimensional coupling of electron cyclotron drift instability and ion–ion two stream instability

Electron cyclotron drift instability (ECDI) and ion–ion two stream instability (IITSI) are both kinetic instabilities that can be present in low-temperature, partially magnetized plasmas. The coupling of instabilities in a three-dimensional configuration leads to the existence of more than one unstable roots to the kinetic dispersion relation. In this paper, a generalized method has been developed for numerically evaluating solutions to the three-dimensional dispersion relation for coupled ECDI and IITSI, assuming cold singly and doubly charged ions and a Maxwellian velocity distribution function for the electrons. The present study demonstrates the coupling between ECDI and IITSI that affects the most unstable mode as a function of the wavenumbers in three dimensions and various plasma properties, including the applied electric field, magnetic field, electron temperature, ion velocities, and plasma density. One of the most notable results is that, while the most unstable mode with the largest growth rate is in the direction of the E×B drift in the two-dimensional cases, the most unstable mode for the three-dimensional configuration occurs in the oblique direction between the applied electric field and the E×B drift. This agrees with experimental observations in cross field plasma sources using coherent Thomson scattering.

Retraction Note: Proof of the Jacobian Conjecture in the Two-Dimensional Case and Global Isochronous Centers of Polynomial Hamiltonian Differential Systems

This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1134/S001226612303014X

The “most classical” states of Euclidean invariant elementary quantum mechanical systems

Complex techniques of general relativity are used to determine all the states in two- and three-dimensional momentum spaces in which the equality holds in uncertainty relations for non-commuting basic observables of Euclidean invariant elementary quantum mechanical systems, even with non-zero intrinsic spin. It is shown that while there is a 1-parameter family of such states for any two components of the angular momentum vector operator with any angle between them, such states exist for a component of the linear and angular momenta only if these components are orthogonal to each other, and hence, the problem is reduced to the two-dimensional Euclidean invariant case. We also show that analogous states exist for a component of the linear momentum and of the center-of-mass vector only if the angle between them is zero or an acute angle. No such state (represented by a square integrable and differentiable wave function) can exist for any pair of components of the center-of-mass vector operator. Therefore, the existence of such states depends not only on the Lie algebra but on the choice of its generators as well.

DETECTION OF SOURCE TERM IN AN ABSTRACT FRACTIONAL SUBDIVISION MODEL BY THE MODIFIED QUASI-BOUNDARY VALUE METHOD WITH A PRIORI AND A POSTERIORI ESTIMATE

The paper discusses the inverse problem of determining an unknown source term in a fractional elliptic equation from measured internal data. In order to solve the considered problem, a modified quasi-boundary value method is used. Applying this method a regularized solution is constructed. An a priori and a posteriori error estimates between the exact solu- tion and its regularized approximation are obtained. A brief analysis is conducted to clarify the relationship between the suggested regularization method and several standard methods. Moreover, numerical results are presented in the one-dimensional case and two-dimensional case to illustrate the accuracy and efficiency of this method.